Lie Algebras
Learn the fundamental objects in algebra, especially representation theory, with hands-on experience in computing representations and constructing sophisticated proofs for some powerful (and quite beautiful!) results. Practice focuses on the basic structures of simple Lie algebras over the complex numbers, as well as the theory of highest weight representation including Verma modules and enveloping algebras, concluding with Weyl's character formula for finite-dimensional simple modules. Additional topics include root systems, Cartan subalgebras, Cartan/triangular decomposition, Dynkin diagrams, and the Killing form.
This is an alternating years course.
A solid grasp of undergraduate linear algebra, as well as experience following long proofs and constructing your own proofs. Students must be very comfortable with proofs in order to understand the material in this course and complete the homework questions adequately. If you are unsure, please discuss this further with your academic mentor. Some prior knowledge of the representation theory of finite groups will also be helpful when grappling with analogous results for Lie algebras, but it is not completely necessary.